Determination of the energy loss and the operating temperatures of a tire is extremely important when considering tire durability. However, the thermomechanical analysis of a rolling tire is a highly complex process due to the effects of temperature on both the mechanical state and the viscoelastic energy dissipation in the tire. (It should be understood that it is not the physical tire itself that is analyzed, but rather the design of the tire, such as would be represented on a CAD/CAM system.). Thus, the computation of the tire temperature field, especially when considering transient heat build-up, is complicated by the temperature dependence of elastic and viscoelastic material properties (thermomechanical coupling).
Modeling the coupled thermomechanical behavior of a steady state rolling tire typically requires that the deformation, energy dissipation, and temperature be determined iteratively. As a result, a steady-state analysis involves updating the temperature dependent elastic and viscoelastic properties as the solution proceeds. The computation usually involves three modules: a deformation module, a dissipation module, and a thermal module. The process is further complicated in a transient analysis where temperature dependent material properties need to be updated at multiple intervals in time.
An iterative process employing finite element analysis (FEA) for updating the material properties for calculating steady state temperatures for a rolling pneumatic tire is described below, and involves an “inner loop” (energy dissipation) computation that iteratively updates the rubber loss modulus G″ for temperature and an “outer loop” (structural loop) computation that updates the storage modulus G′ for temperature. This process would be repeated at regular time intervals in a transient thermal analysis, resulting in a high degree of computation complexity.
FIG. 1 illustrates a prior art technique for analyzing coupled heat generation. The technique is based on the process discussed by researchers at General Motors, as discussed in Whicker, et al. “Thermomechanical Approach to Tire Power Loss Modeling”, Tire Science & Technology, 3, Vol. 9, No. 1, 1981.
The overall process 100 includes three “modules”—a structural analysis module 102, an energy dissipation module 104, and a thermal analysis module 106. The structural analysis module 102 and the thermal analysis module 106 are typically based on commercial finite element analysis (FEA) software, such as ABAQUS. ABAQUS is a well-known suite of general-purpose, nonlinear finite element analysis (FEA) programs, which is used for stress, heat transfer, and other types of analysis in mechanical, structural, civil, biomedical, and related engineering applications. The energy dissipation (EDISS) module 104 is suitably based on software discussed in Ebbott, et al., “Tire Temperature and Rolling Resistance Prediction with Finite Element Analysis,” Tire Science and Technology, 2, Vol. 27, No. 1, 1999.
Assuming that a fully-coupled analysis is to be performed, the analysis starts with the ABAQUS structural tire model 102, where elastic material properties are defined as a function of temperature. Temperatures are initialized and the tire is inflated and load deflected, resulting in a set of element strains (γ).
The strains are transferred to the energy dissipation module (EDISS program) 104. Within the energy dissipation module 104, the loss modulus G″ is also defined as a function of temperature. The temperature is initially set to a constant value. The energy dissipation module 104 calculates energy dissipation for each ring of elements, as well as total energy loss and the resulting tire rolling resistance force. A set of element heat fluxes corresponding to each ring of elements is produced.
In order to obtain tire temperatures, the ABAQUS axisymmetric thermal model 106 is run. The model reads the heat fluxes computed from the energy dissipation module 104, applies appropriate thermal boundary conditions, and computes the tire temperature profile. Completion of this step is considered the end of a typical “uncoupled” open loop analysis.
For further solution refinement a “partially coupled” calculation is used, where the current temperatures (T) output by the thermal analysis module 106 are recycled (looped back) 108 into the energy dissipation module 104 to update the loss modulus (G″). New heat fluxes are calculated, which are then passed to the thermal analysis module 104 for new temperatures, and so on. This process is termed the “energy dissipation” loop—alternatively, the “inner loop”. The energy dissipation loop (104, 106, 108) ends (is “converged”) when two successive temperature calculations (Tn, Tn+1) are within a specified “inner” loop temperature tolerance (ΔTi), which typically requires fewer than ten energy dissipation loop cycles.
Still further solution refinement may be obtained by using a “structural loop” calculation—alternatively termed the “outer loop”. Here, the temperatures from the last converged energy dissipation (inner) loop cycle are re-applied via 110 to the ABAQUS structural analysis module 102 to update the elastic properties for the new temperatures. The process proceeds once again through another energy dissipation loop cycle 108, and so on. The structural loop 110 ends (is converged) when two successive temperature calculations (Tm, Tm+1) are within a specified “outer loop” temperature tolerance (ΔTo), which typically requires about three structural loop cycles.
The “inner loop” computation is readily automated. Since the inner loop involves only EDISS and the ABAQUS thermal wedge model, this phase of the computation is relatively fast. However, for each outer loop cycle, the structural analysis must be rerun with updated elastic properties. This phase of the analysis is computationally slow. Adapting the procedure to transient analyses would potentially require inner and outer loop computations at multiple time steps, increasing the computation time even further.
The method described with respect to FIG. 1 is iterative, and therefore takes time to converge, even more so as the number of discrete elements in the FEA structural analysis is increased.